Attractors in nonlinear dynamical systems are used to represent long term dynamics, which are the most essential feature of nonlinear dynamical systems.^[1,2] The set of initial conditions that can lead to an attractor is defined as its basin of attraction. The separations among basins are the basin boundaries. For a conventional stable attractor, the attractor is enclosed in its basin. We thus can quantify the strength of an attractor by the basin size which is the closest distance between the attractor and its basin boundaries. Typically, the basin boundaries are the stable manifold of saddle points. However, for hidden attractors, their corresponding basins do not intersect with small neighborhoods of any equilibrium points.^[3–6]

Another interesting dynamic is meta-stable states, the system can stay near the vicinity of the meta-stable states for a relatively long time. Meta-stable dynamics is usually due to the appearance of saddles, and the switching dynamics among meta-stable states is regarded as useful for information processing.^[7] Usually attractors and meta-stable states are two quite different dynamical states. However, unstable attractors are saddles but with positive measures of basins.^[8–10] Hence this type of attractor is defined in the sense of Milnor attractors.^[11] For these attractors, almost all nearby points within a neighborhood will leave the attractors, leading to a zero basin size. But the remote region in phase space could collapse into the stable manifold of saddles.^[12] Another interesting type of attractor, i.e., partially unstable attractors, with both stable and unstable points can also appear in pulse-coupled oscillators under periodic forcing.^[13]

Unstable attractors are observed in pulse-coupled oscillators with delays, where the delays account for the transmission time of the pulses.^[14–19] Such models can be used to study various natural phenomena, such as neural dynamics where pulses correspond to the generations of spikes, and synchronization of fireflies where pulses are the flashes. The interactions among oscillators are mediated by the pulses. The generation and arrival of pulses are then important events. We can introduce symbolic events based on the related events, which are useful for understanding the unstable attractors.^[20,21]

One typically application of these attractors is used to generate switching dynamics among these states.^[7,22–26] The switching dynamics built upon unstable attractors is sensitive to infinitesimal perturbations. Thus any small noise can induce switching phenomena. One desirable feature is that switching dynamics is only sensitive to finite size perturbations. Thus how to explicitly manipulate these unstable attractors is interesting and important to the applications built upon these novel saddles. In this paper, we present a method to control unstable attractors towards attractors with controllable basin sizes. We first identify the local expanding dynamics and then introduce the local contracting dynamics. The basin sizes and staying time can be controlled by the cooperation of expanding and contracting dynamics.

This paper is organized as follows. In Section 2, the model of pulse-coupled oscillators is described, and also the evolution rate for the local expanding dynamics is derived. In Section 3, we introduce the contracting dynamics, investigate the local dynamic behavior, and derive the condition of the appearance of contracting dynamics. In Section 4, we investigate the staying time and attractors with controllable basin sizes, and we realize the switching dynamics that are only sensitive to perturbations larger than the basin sizes. Finally, the conclusion and discussion are presented.

The effects of pulses are expanding for the active firing oscillators under the current function, in the sense that the phase difference for any two active firing oscillators increases. In order to manipulate or control these unstable attractors, an intuitive way is to introduce the contracting dynamics so that the arrival of certain pulses could decrease the phase differences among the active firing oscillators. Thus we slightly modified the function U_b(x) in Eq. (3), which mediates the pulse interactions among the oscillators.

Intuitively, the introduction of contracting dynamics is possible. For example, we can make the function in Eq. (3) behave close to a horizontal line in a certain interval where a pulse is received. Then the phase response of two oscillators at slightly different states can actually become close to each other, and hence the corresponding phase difference decreases. Then we consider which interval we should choose to modify the function. To this end, we consider the state of the active firing oscillators. We modify the effect of pulses from these active firing oscillators. Just before the arrival of these pulses, the states of the active firing oscillators are near τ. This is typically true when τ is not relatively large.^[21]

Thus we slightly modify the function in the interval [τ −c, τ + c], where c determines the width of the interval and should be small enough. In this paper, we choose c = 10⁻⁴ and c = 10⁻⁵ as two typical small values. For simplicity, we still use the same form of function U_b(x) but with different parameters within this interval. To be distinguishable, the parameter is denoted by B. The function in the interval [0,1] is continuous. We will show later that c is directly related to the controlled basin sizes.

Thus the new function has two parameters b and B, which is given by 10 where x₁ = τ −c, x₂ = τ + c, y₁ = U_b(x₁), and y₂ = U_b(x₂).

When |B| is sufficiently large, the above function in the small interval [x₁, x₂] becomes horizontal. Then upon the arrival of a pulse, the states of the oscillators within this interval can become closer to each other. Hence contracting dynamics occurs. Other forms of functions which behave horizontally in the interval [x₁, x₂] can also introduce contracting dynamics and hence can also work. Figure 2 shows an example of the modified function.

Fig. 2.

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	Fig. 2. The function by modifying the function in the interval [τ −c, τ + c]. The arrival of a pulse with strength ε′ in this interval can induce contracting dynamics in the sense that the phase difference of two active firing oscillators decreases, while other pulses can induce expanding dynamics.

Here we first investigate how to choose the parameter c which determines the interval for the modified function. The value of c should be small enough, such that global dynamics of the system is not much disturbed. In the following, we analyze the detailed condition for c, with the consideration of simplification on the analysis.

Upon the arrival of a pulse from one active firing oscillator, the state of one of the other oscillators is ϕ, satisfying τ − c < ϕ < τ + c. Then suppose that the arrived pulse is of strength ε′. According to the response process, we need to consider the value . The value of y determines which part of the inverse function we should use. If y > U_b(τ + c), we should apply the inverse function of U_b(y). Otherwise, we should apply the inverse function . When c is much smaller than ε′, the first case is always satisfied. This process is shown in Fig. 2.

Then the arrival of a pulse with strength ε′ for an oscillator with state ϕ could induce a phase change as The right-hand side of the above equation is frequently applied later, and is defined as a new function 11

We then choose two active firing oscillators, and study their evolution of phase difference under the new function. Here we first consider the effect of pulses from the active firing oscillators on the phase difference. The two oscillators are p and q, and assume that p fires first. Then we encounter a piece of events S_p −S_q −R_p −R_q. Just after the event S_q, the states for p and q are x and 0, respectively, with x close to zero. Thus the phase difference between the two oscillators is x. We then calculate the phase difference just after the event R_q.For oscillator q, it receive a pulse from p (i.e., event R_p) with phase τ −x. After x time, the pulse from itself is received (i.e., event R_q), its phase will increase further by x due to the free evolution Eq. (1). Then the phase of oscillator q is For oscillator p, the phase at event R_q can be obtain similarly, which is Then we find that the phase difference is changed from x to ϕ_p − ϕ_q. We obtain the rate of phase change, , due to the arrival of the pulses from themselves, which is 12

Here x is the initial small phase difference for the two active firing oscillators. The depends on the initial phase difference x. We find that this function is monotonous on x. The minimum value is obtained when x is near zero. Thus, 13 By using equation (13) becomes 14

The quantifies the change of phase difference due to the pulses from themselves. If it becomes less than 1, then the effect becomes contracting.

Then we consider the total effect of pulses on the evolution rate during one time period. Again, the contributions can be classified into two parts. One contribution comes from the active firing oscillators as described in Eq. (12). The other contribution comes from the non-active firing oscillators. As the active firing oscillators receive the pulses at the state outside the interval [τ − c, τ + c], then the total effect of these pulses is the same as R_o in Eq. (7). Then the total rate after one time period is 15 When the initial phase difference is close to zero, the evolution rate is minimal given by 16

The critical condition is , which is shown in Fig. 3. The value of is symmetrical with respect to B. In the middle region II, the value of is always larger than 1, indicating expanding behaviors. Then if we increase |B|, decreases. In the case , the phase difference actually decreases after perturbations, and contracting dynamics dominates. In this case, the unstable attractor becomes a stable attractor, we analyze the basin size in the following subsection. The critical value of B is determined by .

Fig. 3.

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	Fig. 3. Regions I and II are the stable and unstable regions, respectively, which are separated by dashed lines obtained according to the critical condition . The parameters are N = 15, b = 3, ε = 0.1, τ = 0.1, and c = 10−5.

In the case that , the unstable attractor becomes a stable attractor. Here we investigate the controlled basin size d. As shown in Fig. 3, region I corresponds to the case with . The range of x in region I determines the basin size. Thus the basin size grows with the increase in |B|, and approaches c. Thus for large |B|, d ≈ c. This can also be understood intuitively as follows. If d >c, the phase for an oscillator could be τ + d or τ −d. Note that the chosen modified interval is [τ−c, τ + c]. Then the arrival of pulses from the active firing oscillators falls out of this region.

Then we estimate the basin sizes. Here we study the dynamics in the return map with one oscillator as reference. Just after the reset of the reference oscillator, we add perturbations. The order of events sequence also matters. In the above analysis, we assume that the pulses from the active firing oscillators are received first. In the case that they are received last, the phase difference between the two active firing oscillators increases first due to the arrival of pulses from the other oscillators. Thus the initial phase difference x is increased to x × R_o which should be smaller than xexp(εb). Thus we can approximate the basin size as 17 where |B| is much larger than its critical value.

We verify the above results by using N = 4 globally coupled oscillators with fixed parameters b = 3 and ε = 0.15. Figure 4(a) shows the controlled basin sizes for the unstable attractors with the parameter B and τ = 0.18. For each B, 100 randomly chosen initial conditions are used to locate the attractors. Here the critical value of B_cr is 2.07. When B is smaller than B_cr, the original unstable attractors are still unstable with a basin size of zero. When B becomes larger than B_cr, the basin sizes become larger than zero and approaches c = 10⁻⁵. In this case, the basin sizes will finally fall within the predicted values of Eq. (17).

Fig. 4.

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Fig. 4. The controlled basin sizes for N = 4 globally coupled oscillators with fixed parameters b = 3 and c = 10−5. (a) The dependence of the basin sizes on the parameter B with fixed parameters ε = 0.15 and τ = 0.18, where the critical value of B is 2.07 indicated by the dashed vertical line. (b) The dependence of the basin sizes on the parameter τ with fixed parameters ε = 0.15 and B = 15.0. Here the shaped region denotes the basin size predicted by Eq. (17) when B is much larger than the critical value.

Furthermore, we choose a larger value of B = 15 and let τ be a variable. Then we uniformly choose 1000 values of τ from the interval [0.17,0.21], where unstable attractors prevail. For each τ, 100 randomly chosen initial conditions are used to locate the attractors. If the attractors are of a structure similar to Eq. (5), we then obtain the basin size. In Fig. 4(b), the dependence of the basin sizes on the parameter τ is shown. Again, we can see that the basin sizes are well predicted by Eq. (17). As long as the value of B is much larger than the critical value, the contracting dynamics dominates, and hence the controlled basin sizes can be predictable. Additionally, the method does not depend on the system parameters as demonstrated in Fig. 4(b).

Then we show a detailed example in Fig. 5 for a relatively higher system with N = 15 globally coupled oscillators. The parameters are chosen as b = 3, ε = 0.2, τ = 0.1, B = 5, and c = 10⁻⁴. Under the current parameter setting, we can locate an attractor with oscillators 6 and 7 as the simultaneous active firing oscillators, which should be an unstable attractor without control. Then we change the phase of oscillators 6 and 7 by ϕ₆ = ϕ₆ + Δϕ₆ and ϕ₇ = ϕ₇ + Δϕ₇. We can see that the system is stable to small perturbations, indicated by the region enclosed by the dashed lines. The dashed line is predicted by ϕ₆ −ϕ₇ = x_cr and ϕ₆ − ϕ₇ = −x_cr, where x_cr = c/exp(0.2×3.0) = 5.5×10⁻⁵.

Fig. 5.

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Fig. 5. An original unstable attractor with oscillators 6 and 7 as the active firing oscillators is successfully controlled to be a stable attractor marked by “+”. The red region denotes its basin, where the basin size is correctly predicted as shown by the dashed line. The nearby point within the basin leads to the stable attractor as shown in green ◀, while the nearby point outside the basin leaves the stable attractor as shown in yellow ⋆. The parameters are chosen as N = 15, b = 3, ε = 0.2, B = 5, and c = 10−4.

Here we first consider the case with . In this case, the local dynamics is unstable. The appearance of contracting dynamics can make approach 1, which can increase the staying time. Thus we can use B to control the staying time. As an example, we choose the global network with N = 15. For B > 0, the critical value B_cr = 1.4218 for ε = 0.1, τ = 0.1, and b = 3. Figures 6(a)–6(c) show the staying time for an unstable attractor with different parameter B, each with the same initial condition. We can see that the staying time increases when B approaching B_cr.

Fig. 6.

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Fig. 6. (a)–(c) The staying time for an unstable attractor is controlled. The staying time (highlighted in gray) becomes longer by decreasing B. (d) For the leaving process in (c), the evolution of phase difference for two active oscillators with time n reveals two processes with two different evolution rates as predicted. The parameters are N = 15, b = 3, ε = 0.1, τ = 0.1, and c = 10−4. The time n denotes the n-th reset of the reference oscillator 1.

We then investigate in detail the leaving process. According to Eq. (16), when the phase difference is much smaller than c, the change rate can be approximated by 18 By plugging the parameters associated with Fig. 6(c) into the above equation, i.e., N = 15,b = 3,ε = 0.1,c = 10⁻⁴, and B = 1.39, we can obtain L₁ = 1.014. Then the phase difference is increased, becoming larger than c. The evolution rate becomes close to that given in Eq. (9), which is 19 By plugging the parameters associated with Fig. 6(c) into the above equation, i.e., N = 15, b = 3, ε = 0.1, c = 10⁻⁴, and B = 1.39, we can obtain L₂ = 1.378. In Fig. 6(d), the evolution of the phase difference is plotted with time n. We can observe two stages of leaving process.

In the case of , the unstable attractors become stable with basin size predicted by Eq. (17) for large enough |B|. Here the system cannot leave the attractor for small perturbations which fall into the basin. However, large perturbations can bypass the basin size and make the system leave the attractor. In such cases, the leaving process manifests as a single process and is described by Eq. (19).

One typical application of unstable attractors is to generate switching dynamics, which can reflect the input information. The switching among unstable attractors is sensitive to infinitesimally small perturbations. Hence the switching dynamics is not robust to typically unwanted small noise.

Here we show that the controlled attractors with predictable basin sizes can be used to generate switching dynamics that are insensitive to small perturbations. This can be achieved by making the basin sizes larger than the strength of small perturbations. The perturbed trajectories are still within the basin, hence the switching phenomenon cannot occur. However, in the presence of larger perturbations, the system can bypass the controlled small basin size, and generate switching dynamics.

As an example, we consider the system of N = 15 oscillators, with c = 10⁻⁴, B = −2.0, ε = 0.1, and b = 3. The basin sizes of the controlled unstable attractors are of the order c. We first introduce small random perturbations with the strength 0.1c on each oscillator at the three arrow positions. As shown in Fig. 7(a), there is no switching phenomenon under these small perturbations. In order to generate switching dynamics, sufficient strong perturbations should be introduced. Thus we consider the random perturbations with the strength 5c, such that the system can bypass the controlled interval. As shown in Fig. 7(b), switching dynamics among three metastable states can be observed. This also implies that the three corresponding stable attractors arise from the three controlled unstable attractors. Thus the basin sizes are of the order c. Under the large perturbations, the system can bypass such small basins and generate switching dynamics.

Fig. 7.

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	Fig. 7. (a) Controlled unstable attractors with basin sizes of the order c are insensitive to small perturbations such as of the order 0.1c, which are introduced at the arrow positions. (b) The strong perturbations of the strength 5c can induce subsequent switching among different attractors. The parameters are chosen as N = 15, b = 3, ε = 0.1, τ = 0.1, B = 2, and c = 10−4.

In summary, we study the control of unstable attractors. We first identify the local expanding dynamics for these attractors. In order to control them, we thus slightly modified the function governing the pulse interactions among the oscillators, with the goal to introduce contracting dynamics. Thus we can manipulate the properties of the unstable attractors through the cooperation of expanding and contracting dynamics.

We then study the local dynamics with both expanding and contracting dynamics. We derive evolution rate of phase difference between two active firing oscillators. When the rate becomes less than one, the local dynamics of the original unstable attractor actually becomes contracting. Thus the original unstable attractor is controlled to be a stable attractor. The controlled basin size is directly related to the width of the modified interval. Besides, we show that such controlled attractors are useful in generating switching dynamics that is only sensitive to large perturbations, and is insensitive to smaller perturbations falling within the small basins. Thus this approach can be used to process information with switching dynamics where noises or disturbances are unavoidable.

Here the presented method can only control unstable attractors with certain types of symbolic events. How all unstable attractors with various event structures can be controlled is still an open question. Similar to the method here, we need to consider the state of active firing oscillators at the arrival of pulses from themselves, as these oscillators are the source of unstable local dynamics. Furthermore, we need to study the effect of the split of a large number of simultaneously active firing oscillators, which means that the unstable local dynamics occurs in a high dimensional phase space.